876 |
Consider the lines ; The shortest distance between L1 and L2 is a) 0 b) c) d)
Consider the lines ; The shortest distance between L1 and L2 is a) 0 b) c) d)
|
IIT 2008 |
|
877 |
Let ABCD is the base of parallelopiped T and Aʹ.BʹCʹDʹ be the upper face. The parallelopiped is compressed so that the vertex Aʹ shifts to Aʹʹ on a parallelepiped S. If the volume of the new parallelopiped is 90% of the parallelopiped T, prove that the locus of Aʹʹ is a plane.
Let ABCD is the base of parallelopiped T and Aʹ.BʹCʹDʹ be the upper face. The parallelopiped is compressed so that the vertex Aʹ shifts to Aʹʹ on a parallelepiped S. If the volume of the new parallelopiped is 90% of the parallelopiped T, prove that the locus of Aʹʹ is a plane.
|
IIT 2004 |
|
878 |
The differential equation representing the family of curves where c is a positive parameter, is of a) Order 1 b) Order 2 c) Degree 3 d) Degree 4
The differential equation representing the family of curves where c is a positive parameter, is of a) Order 1 b) Order 2 c) Degree 3 d) Degree 4
|
IIT 1999 |
|
879 |
A normal is drawn at a point of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is
A normal is drawn at a point of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is
|
IIT 1994 |
|
880 |
Area bounded by and
Area bounded by and
|
IIT 2006 |
|
881 |
Let a, b, c be real numbers. Then the following system of equations in x, y, z + − = 1 − + = 1 − + + = 1 has a) No solution b) Unique solution c) Infinitely many solutions d) Finitely many solutions
Let a, b, c be real numbers. Then the following system of equations in x, y, z + − = 1 − + = 1 − + + = 1 has a) No solution b) Unique solution c) Infinitely many solutions d) Finitely many solutions
|
IIT 1995 |
|
882 |
The domain of definition of the function is a) excluding b) [0, 1] excluding 0.5 c) excluding x = 0 d) None of these
The domain of definition of the function is a) excluding b) [0, 1] excluding 0.5 c) excluding x = 0 d) None of these
|
IIT 1983 |
|
883 |
If then a) b) c) d) f and g cannot be determined
If then a) b) c) d) f and g cannot be determined
|
IIT 1998 |
|
884 |
If f : [1, ∞) → [2, ∞) is given by then equals a) b) c) d)
If f : [1, ∞) → [2, ∞) is given by then equals a) b) c) d)
|
IIT 2001 |
|
885 |
Multiple choices with one or more than one correct answers then a) x = f(y) b) f(1) = 3 c) y increases with x for x < 1 d) f is a rational function of x
Multiple choices with one or more than one correct answers then a) x = f(y) b) f(1) = 3 c) y increases with x for x < 1 d) f is a rational function of x
|
IIT 1984 |
|
886 |
Given and f(x) = cosx – x(x + 1). Find the range of f (A).
Given and f(x) = cosx – x(x + 1). Find the range of f (A).
|
IIT 1980 |
|
887 |
Show that the value of wherever defined, never lies between and 3.
Show that the value of wherever defined, never lies between and 3.
|
IIT 1992 |
|
888 |
Let where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.
Let where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.
|
IIT 1998 |
|
889 |
The values of lies in the interval . . .
The values of lies in the interval . . .
|
IIT 1983 |
|
890 |
If and then (gof)(x) is equal to
If and then (gof)(x) is equal to
|
IIT 1996 |
|
891 |
If 0 < x < 1, then is equal to
If 0 < x < 1, then is equal to
|
IIT 2008 |
|
892 |
Let where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1. a) b) c) d)
Let where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1. a) b) c) d)
|
IIT 1997 |
|
893 |
A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve. a) x2y = 1 b) x = y c) xy = 1 d) x2 = y
A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve. a) x2y = 1 b) x = y c) xy = 1 d) x2 = y
|
IIT 1998 |
|
894 |
For a positive integer n, define then a) a(100) ≤ 100 b) a(100) > 100 c) a(200) ≤ 100 d) a(200) > 100
For a positive integer n, define then a) a(100) ≤ 100 b) a(100) > 100 c) a(200) ≤ 100 d) a(200) > 100
|
IIT 1999 |
|
895 |
Let –1 ≤ p ≤ 1. Show that the equation 4x3 – 3x – p = 0 has a unique root in the interval and identify it. a) p b) p/3 c) d)
Let –1 ≤ p ≤ 1. Show that the equation 4x3 – 3x – p = 0 has a unique root in the interval and identify it. a) p b) p/3 c) d)
|
IIT 2001 |
|
896 |
Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.
Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.
|
IIT 1999 |
|
897 |
If p is a natural number then prove that pn + 1 + (p + 1)2n – 1 is divisible by p2 + p + 1 for every positive integer n.
If p is a natural number then prove that pn + 1 + (p + 1)2n – 1 is divisible by p2 + p + 1 for every positive integer n.
|
IIT 1984 |
|
898 |
Prove by mathematical induction that for every positive integer n.
Prove by mathematical induction that for every positive integer n.
|
IIT 1987 |
|
899 |
Prove that is an integer for every positive integer.
Prove that is an integer for every positive integer.
|
IIT 1990 |
|
900 |
Let a hyperbola pass through the foci of the ellipse . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then a) Equation of the hyperbola is b) Equation of the hyperbola is c) Focus of the hyperbola is (5, 0) d) Vertex of the hyperbola is
Let a hyperbola pass through the foci of the ellipse . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then a) Equation of the hyperbola is b) Equation of the hyperbola is c) Focus of the hyperbola is (5, 0) d) Vertex of the hyperbola is
|
IIT 2006 |
|